1. Hirophysics.com Some Aspects of Numerical Solutions for Partial Differential Equations Austin Andries University of Southern Mississippi Dr. Hironori Shimoyama May 5, 2011

2. Hirophysics.com Laplace’s Equation Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as: Laplace's equation are all harmonic functions and are important in many fields of science including electromagnetism, astronomy, and fluid dynamics. Where is the Laplace operator and is a scalar function. =

3. Hirophysics.com The solution of Laplace’s equation under one of the boundary conditions

4. Hirophysics.com Poisson’s Equation Electrostatics is the posing and solving of problems that are described by the Poisson equation. Finding φ for some given ƒ is an important practical problem, since this is the usual way to find the electric potential for a given charge distribution. If ƒ vanishes to zero then the express becomes Laplace’s equation.

5. Hirophysics.com The case of two unlike point charges

6. Hirophysics.com Multiple point charges

7. Hirophysics.com Solitons A soliton is a self-reinforcing solitary wave (a wave packet or pulse) that travels with constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. They arise as the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems. The blue line is the carrier waves, while the red line is the envelope soliton.

8. Hirophysics.com Examples of solitons

9. Hirophysics.com Interaction of soliton waves

10. Hirophysics.com Future Research Implementation of a higher order of finite difference methods Investigating some of pitfalls with the simple algorithm of finite difference

11. Hirophysics.com Work Cited: Images: http://en.wikipedia.org/wiki/Laplace%27s_equation (slide 2 ) http://en.wikipedia.org/wiki/Poisson%27s_equation (slide 7) http://en.wikipedia.org/wiki/Poisson%27s_equation (slide 18). Code referred to Computational Physics by Landau and Paez

12. Hirophysics.com Appendix: Young-Laplace Equation Soap Films If the pressure difference is zero, as in a soap film without gravity, the interface will assume the shape of a minimal surface. Young–Laplace equation is a nonlinear partial differential equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension. Δp = pressure difference across the fluid interface γ = surface tension, is the unit normal pointing out of the surface, H = is the mean curvature R 1, R 2 = are the principal radii of curvature.

13. Hirophysics.com